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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvrelsymb | Structured version Visualization version GIF version |
Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised and distinct variable conditions removed by Peter Mazsa, 2-Jun-2019.) |
Ref | Expression |
---|---|
eqvrelsymb.1 | ⊢ (𝜑 → EqvRel 𝑅) |
Ref | Expression |
---|---|
eqvrelsymb | ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvrelsymb.1 | . . . 4 ⊢ (𝜑 → EqvRel 𝑅) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → EqvRel 𝑅) |
3 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵) | |
4 | 2, 3 | eqvrelsym 35720 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵𝑅𝐴) |
5 | 1 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → EqvRel 𝑅) |
6 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐵𝑅𝐴) | |
7 | 5, 6 | eqvrelsym 35720 | . 2 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐵) |
8 | 4, 7 | impbida 797 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 class class class wbr 5057 EqvRel weqvrel 35351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-refrel 35632 df-symrel 35660 df-trrel 35690 df-eqvrel 35700 |
This theorem is referenced by: eqvrelth 35726 |
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